A.5 Systems of Equations

Multiple Choice

What would the solution be to this system of linear equations?

• (0,0)

• (2,2)

• (6,0)

• (0,4)

Multiple Choice

What would the solution be to this system of linear equations?

• (0,-1.5)

• (1, -3)

• (5,0)

• (0,0)

Multiple Select

What is a solution to a system of linear equations?

• The point where both lines intersect

• The point where the y-axis and x-axis meet

• An ordered pair that both lines share

• A salt dissolved in water

Solutions are where two lines intersect on a graph and these are our answers.

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The solution here is (-1,3) because that is where the lines intersect!​

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​Solutions=answers

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Systems of Equations

Systems of equations are used when we are given two pieces of information and asked for two answers from it, like our first question.

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YOU MUST WATCH the video on the next slide to answer the questions

A way of looking at things...

Another way to find out if a set of points, (x,y) is a solution is by looking at the graph.

A Graphical Approach

(One Solution)

Where ever the graphs of linear functions intersect we have a solution.

In this case we have one solution for the system y = -x + 7 and y = 2x + 1 at the point (2, 5).

A Graphical Approach

(No Solutions)

When lines never intersect, then there are no solutions to the system of equations.

Notice in the example that the lines have the same slope and different y-intercepts. (Recall: y=mx+b)

A Graphical Approach

(Infinitely Many Solutions)

Notice in the example that the equations look different at first, but after a little algebra, we see that they are the same line. There are infinitely many solutions.

Multiple Choice

How many solutions will this system have? 

• No solution

• One Solution

• I Don't Know

• Infinitely Many Solutions

Multiple Choice

^{What is the solution?} 

• 1

• -2

• (1, 2)

• (1, -1)

Multiple Choice

How many solutions does this system of equations have?

• One Solution

• No solution

• Infinitely Many Solutions

• Two Solutions

Multiple Choice

If a system of equations has no solution, what does the graph look like? 

• intersecting lines

• parallel lines

• skew lines

• intersecting lines

Multiple Choice

When would we get "infinitely many solutions" to a system of equations?

• The lines are parallel, never intersecting

• The lines cross at one point

• The lines are the same and overlap at all points

• The lines cross at 2 points

Multiple Choice

When will we get "no solutions" from a system of equations?

• The two lines cross once

• The two lines are the same and overlap

• The two lines are parallel and never touch

• The two lines cross at 2 points

Use DESMOS to answer the following questions

Match

Match the following graphs to the systems of equations

 

 

  

   

• $$y=\frac{2}{3}x+1$$  

  $$y=\frac{2}{3}x-2$$  

• $$y=-x+7$$   $$y=2x+1$$  

• $$y=-2x+4$$   $$y=-2x+4$$   

Multiple Choice

Which of the following graphs would fit the system of equations?

$$y=2x-4$$  

$$y=-\frac{1}{2}x+1$$  

Fill in the Blank

What is the solution for x given the graph to the systems of equations

Fill in the Blank

What is the solution for y based on the system:

Fill in the Blank

Please list your answer as a coordinate (x,y)

Fill in the Blank

List the solution as a coordinate (x,y):

Fill in the Blank

Write you answer as a coordinate (x,y).

Fill in the Blank

List the answer as a coordinate (x,y):

Multiple Select

Check all that apply:

• x= -5/2

• y=-5/2

• No Solutions

• Infinitely Many Solutions

• x=0

Multiple Select

Check all that apply:

• x= -5/2

• y=-5/2

• No Solutions

• Infinitely Many Solutions

• x=0

Word Problems with Systems of equations

You must write down and show your work on a piece of paper

You need to write down what x= and y=

More money, more problems

​You recently received some birthday money, $130 to be exact. You count the number of bills you have and find that you have 17 bills, all either $10 or $5 bills.

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How many of each bill do you have?​

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Hint: Just try guessing and checking first!​

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Poll

Which do you think is correct?

• nine $5 bills and eight $10 bills

• eight $5 bills and nine $10 bills

• ten $10 bills and seven $5 bills

• six $10 bills and eleven $5 bills

How does this have to do with math?

Sometimes, in math, we are asked for two answers, like how many $5 bills AND $10 bills. Systems of equations are how we solve these problems. Now use this knowledge to solve the following problems...

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Graphing Systems Word Problems

• Read the text (then re-read the text)

• Write what x= and y=

• Set up two (2) equations

• Graph both equations on DESMOS

• Look for the intersection point

• Answer the question

Multiple Choice

There are 15 animals in a barn.  These animals are horses and chickens.  There are 44 legs in all.  Which system of equations represents the situation?

• x + y = 15 4x + 2y = 44

• 4x + 2y = 15 x + y = 44

• x = 2y + 44 4x = y + 15

• 2x - 4y = 44 x - y = 15

Multiple Choice

Justin is running a concession stand at a soccer game. He sells nachos and sodas. Nachos cost $1.50 each and sodas cost $0.50 each. At the end of the game, Justin made a total of $78.50 and sold a total of 87 nachos and sodas combined. Which system of equations represents this situation?

• (35, 52)

• (18, 69)

• ( 52, 35) 

• (61, 26)

Multiple Choice

Raegan went to the store to buy vegetables. The cost of 5 squash (x) and 2 zucchini (y) was $1.32. Three squash and 1 zucchini cost $0.75.

• $$5x\ +\ 2y\ =\ 1.32$$ $$3x\ +\ y\ =\ .75$$

• $$2x\ +\ 5y\ =\ 1.32$$ $$x\ +\ 3y\ =\ .75$$

• $$5x\ +\ 2y\ =\ .75$$ $$3x\ +\ y\ =\ 1.32$$

• $$2x\ +\ 5y\ =\ 1.32$$ $$x\ +\ y\ =\ .75$$

Multiple Choice

Alexis finds that she can give 3 haircuts and 2 hair dyes in 315 minutes. Giving 2 haircuts and 4 hair dyes takes 450 minutes.  How long does it take her to do a haircut?

3x + 2y = 315

2x + 4y = 450

• 45 minutes

• 90 minutes

• 60 minutes

• 30 minutes

Multiple Choice

A large pizza at Palanzio’s Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. Which system of equations could be used to find the number of toppings when both companies cost the same amount? 

• y = 6.80 + .65x y=7.30+.90x

• x + y = 6.80 x + y = 7.30

• y = 6.80+.90x y = 7.30 + .65x

• y + .90x = 6.80 y + .65x = 7.30

Multiple Choice

On Monday Joe bought 10 cups of coffee and 5 doughnuts for his office at the cost of $16.50.  It turns out that the doughnuts were more popular than the coffee.  On Tuesday he bought 5 cups of coffee and 10 doughnuts for a total of $14.25.  Which equations could be used to determine the cost of the coffee? 

• 10c + 5d = 14.25 5c + 10d = 16.50

• 10c + 5d = 16.50 5c + 10d = 14.25

• c + d = 10 5c + 10d = 16.50

• c + d = 5 5c + 10d = 16.50

Multiple Choice

Dwayne mowed his next door neighbor’s lawn for a handful of dimes and nickels, 80 coins in all.  Upon completing the job he counted out the coins and it came to $6.60.  Which system of equations could be used to find the exact number of dimes and nickels? 

• d + n = 6.60 .10d + .05n = 80

• d + n = 80 d + n = 6.60

• d + n = 80 .10d + .05n = 6.60

• d + n = 80 .05d + .10n = 6.60

Multiple Choice

Last season two running backs on the Steelers football team rushed a combined total of 1550 yards. One rushed 4 times as many yards as the other. Let x and y represent the number of yards each individual player rushed. Which system of equations could be used?

• x + y = 1550

  y = 4x

• x + y = 1550

  y = x + 4

• y - x = 1550

  y = 4x

• y = 1550 + x

  y = x + 4

Multiple Choice

The school that Payton goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Which equations represents the system that could be used?

• 1s + 3c = 38

  2s + 3c = 52

• 3s + 1c = 38

  3s + 2c = 52

• s + c = 38

  s + c = 52

• 3s + 3c = 38

  1s + 2c = 52

Multiple Choice

Michelle and Berenice are selling fruit for a school fundraiser. Customers can buy small boxes of oranges (x) and large boxes of oranges (y). Michelle sold 3 boxes of small oranges and 14 boxes of large oranges for a total of $203. Berenice sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220.

• $$14x\ +\ 3y\ =\ 203$$ $$11x\ +\ 11y\ =\ 220$$

• $$3x\ +\ 14y\ =\ 203$$ $$11x\ +\ 11y\ =\ 220$$

• $$3x\ +\ 14y\ =\ 220$$ $$11x\ +\ 11y\ =\ 203$$

• $$14x\ +\ 3y\ =\ 220$$ $$11x\ +\ 11y\ =\ 203$$

Multiple Choice

Alan decides to call for some fast food delivery. If he chooses McDonalds, the delivery fee is $7 and each sandwich costs $3.99. For KFC, there is no delivery fee but each sandwich (x) is $4.99. Let y be the total cost.

• $$3.99x\ +\ 4.99y\ =\ 7$$   $$x+\ y\ =\ 0$$  

• $$y\ =\ 10.99x$$   $$y\ =\ 4.99x$$  

• $$y\ =\ 4.99x\ +\ 7$$   $$y\ =\ 3.99x$$  

• $$y\ =\ 3.99x\ +\ 7$$   $$y\ =\ 4.99x$$  

Multiple Choice

Maddy sold 10 packages of sugar cookies and 2 packages of oatmeal cookies for $56. Adam sold 9 packages of sugar cookies and 3 packages of oatmeal cookies for $60. What is the price of 1 pack of sugar cookies and 1 pack of oatmeal cookies?

• Sugar Cookies $5  Oatmeal Cookies $3

• Sugar Cookies $4 Oatmeal Cookies $8

• Sugar Cookies $2 Oatmeal Cookies $6

• Sugar Cookies $3 Oatmeal Cookies $6

Multiple Choice

Christi went to the grocery story.  On Monday she purchased 4 apples and 6 bananas for a total of $13.  On Wednesday she purchased 3 apples and 7 bananas for a total of $13.50.  Which system of equations represents the situation?

• 4x + 6y = 3 13.5x - 13y = 6

• x + y = 4 x - y = 6

• 4x + 6y = 13 3x + 7y = 13.5

• 4x - 6y = 13 3x - 7y = 13.5

Multiple Choice

The 8th grade class at Potomac Shores Middle School rented and filled 11 vans and 3 buses with 283 students. The 8th grade class at Potomac Middle School rented and filled 12 vans and 14 buses with 770 students. How many students were on each van and each bus?

• Van 43 Bus 14

• Van 11 Bus 21

• Van 14 Bus 43

• Van 17 Bus 45